Representation Theoretical Methods in the Theory of Special Function

拟议的研究涉及计算机探索和信息传递，将几个数学领域联系起来，尤其是表示论、特殊函数理论和组合学。这些联系提供了一个强大的发现工具，因为在一个领域可能非常明显的结果和机制往往在另一个领域转化为非常不平凡和意想不到的事实。这种类型的活动也非常适合培训年轻的研究人员，并为他们提供机会，以发现在我们的计算机时代可以进行某些研究的方式。组合解释将数学信息--无论是代数信息、分析信息、逻辑信息或其他信息--翻译成视觉信息。在这种背景下，即使是背景有限的学生也可以体验到发现的喜悦。出于这些原因，PI计划继续通过研究生课程和研讨会将所有当前的研究成果带入课堂。事实上，国际和平协会的学生和合作者在国家科学基金会以前的支持下开展的大部分工作都是由这样的课堂活动产生的。代数组合学的子域，也就是该奖项下开展的研究的焦点，是由Pi的表示理论方法创建的，该方法针对Macdonald(1988)关于他现在著名的对称多项式基的猜想。1990年‘S与马克·海曼的合作，得到了用麦克唐纳多项式表示的“对角调和”空间的Frobenius特征的猜想公式。PI和Haiman早期的计算机探索产生了数据，这些数据揭示了对角调和与“停车函数”(一种由计算机科学家创造的五颜六色的组合结构)之间惊人的密切联系。这些发现导致了Haglund等人的提法。这个猜想给出了对角调和函数的Frobenius特性一个漂亮的关于停车函数的显式公式。大约在2000年，马克·海曼用代数几何工具证明了与PI联合提出的最初猜想。这引起了代数几何学家对这一研究领域的关注。这种关注导致了该领域的真正令人惊讶的丰富，出现了各种新工具和未决问题。最近，PI和他的合作者利用这些新的发现，成功地建立了一个无穷无尽的“洗牌猜想”族，将一个完整的对称函数算子李代数与新的“停车函数”类物体联系起来。在这个奖项的支持下，PI计划使用在这一领域20年的努力中开发的大量工具，与他目前的博士生Emily Leven、Yeonkyung Kim和Marino Romero合作解决这些问题。PI和他现在的学生所取得的初步结果是非常有希望的。事实上，最近Lven已经成功地解决了这些新猜想的一个无穷子族。从历史上看，难题一直是基本数学发现的源泉。在这方面，我们的特定调查领域也不应例外。

The proposed research involves computer explorations and the transfer of information connecting several areas of mathematics, most particularly Representation Theory, the Theory Special Functions and Combinatorics. These connections provide a powerful vehicle of discovery, since results and mechanisms which may be quite obvious in one area often translate into highly nontrivial and unexpected facts in one of the other areas. This type of activity is also highly suitable for training young researchers and providing them with the opportunity to discover the manner in which some research can be carried in our Computer Age. Combinatorial interpretations translate mathematical information, be it algebraic, analytical, logical or otherwise, into visual information. Under this setting, even students with limited background can be brought to experience the joy of discovery. For these reasons the PI, plans to continue the practice of bringing all current research efforts right into the classroom through graduate courses and seminars. In fact most of the work of the students and collaborators of the PI carried out under prior NSF support resulted from such classroom activities. The subfield of Algebraic Combinatorics, that is the focus of the research carried out under this award was created by the PI's representation theoretical approach to the (1988) conjectures by Macdonald concerning his now-famous symmetric polynomial basis. The 1990's joint work of the PI and Mark Haiman led to a conjectured formula for the Frobenius characteristic of the space ``Diagonal Harmonics'' in terms of the Macdonald polynomials. Early computer explorations by the PI and Haiman yielded data which revealed a surprisingly intimate connection of Diagonal Harmonics with ``Parking Functions'' (a colorful combinatorial structure created by computer scientists.) These discoveries led to the formulation by Haglund et al. of the Shuffle Conjecture which gives the Frobenius Characteristic of Diagonal Harmonics a beautiful explicit formula in terms of Parking Functions. Around the year 2000, Mark Haiman proved the original conjectures formulated jointly with the PI by Algebraic Geometrical tools. This brought the attention of the Algebraic Geometers to this field of investigation. This attention resulted in a truly surprising enrichment of the field with a variety of new tools and open problems. Very recently the PI and his collaborators, using these new findings succeeded in formulating an infinite family of ``Shuffle Conjectures'' connecting a whole Lie Algebra of Symmetric Function Operators to new ``Parking Function'' like objects. Under the support of this award the PI plans to use the vast collection of tools developed in two decades of efforts in this area to work on these problems in collaboration with his present PhD students Emily Leven, Yeonkyung Kim and Marino Romero. Preliminary results obtained by the PI and his present students have been very promising. In fact, recently Leven has succeeded in solving an infinite subfamily of these new conjectures. Historically, difficult problems have been the source of fundamental mathematical discoveries. Our particular area of investigation should be no exception in this respect.